Measurability, integrability and large algebras

We prove that the family of Lebesgue integrable functions which are nowhere Riemann integrable is strongly 2ᶜ 2 c -algebrable. We show that there is a free algebra A A with c c generators consisting of continuous functions such that for any nonzero g∈ A g ∈ A there is a free algebra Bg B g with 2ᶜ 2 c generators consisting of measurable functions such that f∘ g f ∘ g is nonmeasurable on any interval for any nonzero f∈ Bg f ∈ B g . It turns out that such B B cannot be found for any linear space of continuous g ’s. These give the best possible improvements of results obtained by Azagra, Fernández-Sánchez, Muñoz-Fernández, Rodríguez-Vidanes, Sánchez, Seoane-Sepúlveda and Trutschnig.